Lot's of assumptions here including all of Jeff's, Lowell's that you don't have an abbreviation like COR that matches to Corporation and that also there are not extra parens in Table2, but here's a shot at it:

WITH Table1([AppID],[Appname]) AS (

SELECT 'Null','CSO Services Operations' UNION ALL

SELECT 'Null',' CSP' UNION ALL

SELECT 'Null','AAC Claims Systems'

)

,Table2([AppID],[Appname]) AS (

SELECT '1','Corporate Services Operations(CSO)' UNION ALL

SELECT '2','Credit Servicing Portal(CSP)' UNION ALL

SELECT '3','American Assuarance Company (AAC) Claims Systems'

)

SELECT a.*, e.*

FROM Table2 a

CROSS APPLY dbo.PatternSplitCM(a.Appname, '[()A-Za-z]') b

CROSS APPLY (SELECT RIGHT(Item, LEN(Item)-CHARINDEX('(', Item))) c(Item2)

CROSS APPLY (SELECT '%' + STUFF(Item2, LEN(Item2), 1, '') + '%') d(Item3)

INNER JOIN Table1 e ON e.AppName LIKE Item3

WHERE [Matched]=1 AND CHARINDEX('(', Item) > 0 AND CHARINDEX(')', Item) > 0

PatternSplitCM can be found in the 4th article in my signature links.

I wouldn't want to be responsible for this running in Prod though without a whole lot of real thorough testing through all the possible cases. Still, it might be possible to make it work if you could apply additional assumptions like that the abbreviation always starts in the first non-blank character of Table1.